Tubings of Rooted Trees

Abstract

A tubing of a rooted tree is a broad term for a way to split up the tree into induced connected subtrees. They are useful for computing series expansion coefficients. This thesis discusses two different definitions of tubings, one which helps us understand Dyson- Schwinger equations, and the other which helps us understand the Magnus expansion. Chord diagrams are combinatorial objects that relate points on a circle. We can explicitly map rooted connected chord diagrams to tubings of rooted trees by a bijection, and we explore further combinatorial properties arising from this map. Furthermore, this thesis describes how re-rooting a tubed tree will change the chord diagram. We present an algorithm for finding the new chord diagram by switching some chords around. Finally, a different notion of tubings of rooted trees is introduced, which was originally developed by Mencattini and Quesney [27]. They defined two sub-types of tubings: vertical and horizontal which are used to find coefficients in the Magnus expansion. These two types of tubings have an interesting relationship when the forests are viewed as plane posets.